Bayesian Inference For The Coefficient Identification Problems Of Elliptic Equations

Inverse problems often arise in the fields of non-destructive testing,medical imag-ing,seismic detection,and stealth warplanes,and are the focus and difficulty of domestic and international research.We explore a class of problems of practical importance to parametric inversion-the problem of identifying the coefficients of elliptic equations.In mathematics,people pay attention to the existence,uniqueness and stability of this kind of problems,and a lot of research in this process has laid a solid foundation for the de-velopment of inverse problems.In engineering,the problems in question have a wide application context,attracting a considerable number of researchers and giving rise to a wide variety of numerical algorithms applicable to practical contexts.Both theoretical analysis and numerical algorithms have provided the impetus for the development and application of the field.A notable characteristic of the above problem is not well posed（ill-posed）,i.e.the solution is not continuously dependent on data.In practice,the observed data contain random noise and the unsuitability can cause large perturbations in the solution.In order to obtain a stable solution,stable numerical algorithms need to be designed.The most important class of these algorithms is the Tikhonov regularisation method,which gives an approximate stabilisation of the original problem.We can solve this stabilisation problem to obtain an approximate solution of the original problem.This method has been success-fully applied to a variety of practical engineering problems that are ill-posed.However,the method focuses on stability recovery and ignores the stochastic nature of the data noise.It is difficult to introduce stochasticity into theoretical and algorithmic discussions using this method.For this reason,it is necessary to consider statistical probabilistic means of analysing inverse problems.As a basic probabilistic statistical tool,Bayesian methods have often been used in recent years to deal with inverse problems.Its basic idea is to view the inverse problem as a parametric inference problem,which in turn transforms the solution model of the inverse problem into a computational problem of the posterior distribution.The posterior distribution is decomposed into a likelihood function and a prior distribution,as known from the conditional probability formula.This points us in the direction of solving the inverse problem.For many inverse problems,Bayes method has two problems that need to be clarified:1.The suitability proof of a posterior distribution;2.Approximation or simulation of a posterior distribution.To solve the first problem,three metric tools:Hellinger distance,Wasserstein distance and Kullback-Leibler scatter are used in this paper to prove Hellinger stability,Wasserstein stability and Kullback-Leibler stability in the inverse problem of elliptic equation coefficient identification based on the Bayesian approach,respectively.In order to solve the second problem,this paper uses the Bayesian sampling method based on Markov chain Monte Carlo（MCMC）sampling to compute the coefficients identification of one and two dimensional elliptic equations.It can be seen from the inversion results that the posterior mean and posterior median of the inversion parameters converge to the true value,and the accuracy is above 10^-3.